DART: Improvement of thermal infrared radiative transfer modelling for simulating top of atmosphere radiance

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DART: Improvement of thermal infrared radiative transfer modelling for simulating top of atmosphere

radiance

Ying-Jie Wang, Jean-Philippe Gastellu-Etchegorry

To cite this version:

Ying-Jie Wang, Jean-Philippe Gastellu-Etchegorry. DART: Improvement of thermal infrared radia- tive transfer modelling for simulating top of atmosphere radiance. Remote Sensing of Environment, Elsevier, 2020, 251, pp.112082. �10.1016/j.rse.2020.112082�. �hal-03082738�

1

DART: improvement of thermal infrared radiative transfer

1

modelling for simulating top of atmosphere radiance

2 3

Yingjie WANG

, Jean-Philippe Gastellu-Etchegorry

4

5

1

CESBIO, CNES-CNRS-IRD-UPS, University of Toulouse, 31401 Toulouse CEDEX 09, 6

France 7

8

Correspondence to: Yingjie WANG (yingjiewang1102@gmail.com) 9

10

Abstract

11

12

Land surface temperature (LST) is increasingly needed for studying the functioning of the 13

Earth's surface at local to global scale. Radiative transfer (RT) models that simulate top of 14

atmosphere (TOA) radiance are essential tools to derive accurate LST from thermal infrared 15

(TIR) signals of Earth observation (EO) satellites. DART (Discrete Anisotropic Radiative 16

Transfer) is one of the most accurate and comprehensive three-dimensional models that 17

simulate RT in the Earth-atmosphere system. Up to version 5.7.3, the mean absolute error 18

(MAE) of DART atmospheric TIR radiance of six standard atmospheres (USSTD76, 19

TROPICAL, MIDDLATSUM, MIDDLATWIN, SUBARCSUM, SUBARCWIN) over 3.5 𝜇m 20

- 20 𝜇m was 3.1 K compared to the reference atmospheric RT model MODTRAN, which is 21

much larger than the 1 K accuracy needed by most LST applications. Also, the radiance error 22

reached 2.6 K for some TIR bands whereas the noise equivalent differential temperature (NeDT)

23

2

of satellite TIR sensor is usually less than 0.4 K. Recently, the DART atmospheric RT 24

modelling was greatly improved by (1) introducing the equivalent absorption cross-section of 25

five most absorbing gases (H

O, CO

, O

, CH

, N

O), and (2) implementing a double-layer 26

thermal emission method. The MAE of DART atmospheric TIR radiance of six standard 27

atmospheres and actual atmospheres over France and the Mediterranean Sea is now better than 28

1.0 K. The band radiance error is less than 0.2 K in the EO satellite TIR bands. DART is still 29

accurate if the temperature profiles of standard atmospheres are offset by less than 6 K and if 30

the viewing zenith angle is less than 50°. In short, the improved DART meets the requirements 31

of both LST applications, and present and future TIR EO satellite missions. It is already 32

available to scientists (https://dart.omp.eu).

33 34

Keywords: DART, radiative transfer, atmosphere, thermal infrared, absorption cross-section,

35

MODTRAN 36

37

1. Introduction

38

39

Land surface temperature (LST) has a wide range of applications in different fields:

40

evapotranspiration, soil moisture, precision agriculture, urban climate, river environments, 41

oceanography, etc. (Dugdale, 2016; Khanal et al., 2017; Kilpatrick et al., 2015; Voogt and Oke, 42

2003; Wang et al., 2006; Wang and Qu, 2009). Due to its high temporal resolution, broad 43

coverage and low cost, thermal infrared (TIR) remote sensing is an ideal tool to measure LST 44

(Li et al., 2013). Therefore, an increasing number of space missions embark sensors with TIR 45

bands. For example, the Trishna mission of French Space Agency (CNES) and Indian Space 46

Research Organization (ISRO), to be launched in 2024-2025, will embark a sensor that has four 47

TIR bands with noise equivalent differential temperature (NeDT) of 0.3 K at 300 K (Lagouarde

48

3

et al., 2018). The sea and land surface temperature radiometer on board the European Space 49

Agency (ESA) Sentinel 3 satellite has three TIR bands with NeDT of 0.05 K at 270 K (Donlon 50

et al., 2012). The National Aeronautics and Space Administration (NASA) Landsat 8 satellite 51

has a TIR sensor with NeDT of 0.4 K at 300 K (Irons et al., 2012)

Landsat 9 satellite, due to 52

be launched on 2021, should embark a TIR sensor similar to the Landsat 8 TIR sensor 53

(McCorkel et al., 2018).

54

55

Most LST applications require accuracy less than 1 K (Sobrino et al., 2016). Although the 56

sensibility (NeDT) of most satellite TIR sensors is less than 0.4 K, the LST derived from 57

remotely sensed data is usually less accurate, mainly due to atmospheric conditions, topography, 58

land surface heterogeneity, and directional effects (Bento et al., 2017; Bonafoni, 2016; Ermida 59

et al., 2018; He et al., 2019; Price, 1983; Vermote et al., 2002). Therefore, there is a need to 60

better link LST and observations from satellite TIR sensors. Physical models that accurately 61

simulate TIR radiative transfer (RT) in the Earth-atmosphere system are essential tools.

62

However, most RT models are either for the atmosphere (e.g., 4A/OP, MODTRAN, LBLRTM, 63

RFM, ARTS) or for the Earth surfaces (e.g., Rayspread, RAPID3, FLiES, SAIL) (Berk et al., 64

2015; Buehler et al., 2018; Clough et al., 2005; Huang, 2018; Kobayashi and Iwabuchi, 2008;

65

Scott, 1974; Verhoef, 1984; Vincent and Dudhia, 2017; Widlowski et al., 2006). DART 66

(Discrete Anisotropic Radiative Transfer) is one of the few models that simulate RT in the 67

Earth-atmosphere system. Its Earth surface RT modelling accuracy in the short and long waves 68

was already verified in the four phases of the RAdiative transfer Model Intercomparison (RAMI) 69

project (Widlowski et al., 2015, 2013, 2007) and in experiments (Guillevic et al., 2003, 2013).

70

Its atmospheric short wave RT modelling was validated with the reference atmospheric RT 71

model MODTRAN-5 (Gastellu-Etchegorry et al., 2017; Grau and Gastellu-Etchegorry, 2013).

72

4

However, as presented below, its atmospheric RT modelling in the TIR region did not meet the 73

requirements of LST applications and TIR Earth observation (EO) satellite missions.

74 75

DART simulates atmosphere in three altitude regions: (1) the bottom atmosphere inside the 76

Earth scene voxel matrix, (2) the mid-atmosphere made of voxels of any size, and (3) the high 77

atmosphere made of layers. Voxels in the mid-atmosphere allows DART to simulate the spatial 78

heterogeneity of the atmosphere backscattering. Any atmosphere layer is homogeneously filled 79

with gasses, aerosols and/or clouds that have specific physical (i.e., temperature, pressure, 80

density) and spectral (i.e., absorption/scattering extinction coefficient, scattering phase function) 81

properties. The atmospheric RT modelling relies on the spectral application of Beer's law and 82

band mean optical properties (Gastellu-Etchegorry et al., 2004). The extinction coefficient at 83

each layer is calculated so that the use of Beer's law gives the same vertical atmospheric 84

transmittance as MODTRAN, assuming that the cross-section of the gases is independent of 85

pressure and temperature. Although initial methodology simulates accurate atmospheric 86

radiance in short waves, the TIR radiance could differ significantly from MODTRAN. For 87

example, in [3.5 𝜇m - 20 𝜇m] region, its mean absolute error (MAE) of top of atmosphere (TOA) 88

atmospheric TIR brightness temperature (BT) of the USSTD76 atmosphere was 3.1 K, which 89

is much larger than LST application requirements. This is due to three approximations: (1) gas 90

absorption cross-section is independent of pressure and temperature; (2) gas absorption 91

transmittance is computed with Beer's law and band mean optical properties; (3) the method 92

that computes layer thermal emission is only suited to optically thin atmosphere.

93 94

Actually, the gas absorption cross-section varies with pressure and temperature due to the 95

Doppler and Lorentz broadening. Many atmospheric RT models, including MODTRAN, 96

compute the pressure- and temperature-dependent gas absorption cross-section lines based on

97

5

the high resolution (spectral resolution up to 0.001 cm

) spectroscopic databases, such as 98

HITRAN and GEISA (Jacquinet-Husson et al., 2016; Rothman et al., 2009). These models 99

compute the absorption transmittance with either the exact gas absorption cross-section lines 100

(line-by-line models like 4A/OP) or the statistically determined gas absorption cross-section 101

lines (band models like MODTRAN). Therefore, their absorption transmittance is usually more 102

accurate than the absorption transmittance calculated with Beer's law and band mean absorption 103

cross-sections. In addition, the thermal emission of an atmosphere layer computed with the 104

layer mean temperature is incorrect if the layer is optically thick. MODTRAN improved it by 105

computing the layer thermal emission with the "linear-in-optical depth" method (Clough et al., 106

1992).

107 108

This paper presents two major improvements of DART TIR RT modelling in order to meet the 109

requirements of LST applications and TIR EO satellite missions: (1) introduction of the 110

pressure- and temperature-dependent equivalent absorption cross-section of five most 111

absorbing gases (H

O, CO

, O

, CH

, N

O); (2) implementation of an efficient double-layer 112

thermal emission method that is adapted to most atmospheric conditions. Limits of these two 113

improvements are also discussed. Then, the improved DART is compared with MODTRAN-5 114

using standard atmosphere profiles and the actual atmosphere profiles from ECMWF reanalysis 115

dataset.

116 117

2. DART model

118

119

DART (https://dart.omp.eu) has been developed at CESBIO since 1992 (Gastellu-Etchegorry 120

et al., 2017, 1996). It is one of the most accurate and comprehensive 3D RT models for the 121

remote sensing community. It simulates the radiative budget, bi-directional reflectance factor

122

6

(BRF) and images at any altitude and along any viewing direction for 3D natural and urban 123

scenes, with topography and atmosphere (Figure 1), from visible to thermal infrared region. For 124

that, it uses an iterative discrete ordinate method (DOM) that tracks radiation fluxes along a 125

finite number of discrete directions. It also simulates terrestrial and aero-spatial LiDAR signal 126

(point cloud, waveform, photon counting) with an approach that combines Monte Carlo method 127

and DOM.

128

129

Figure 1. DART 3D mock-up and voxel matrix. Atmosphere is separated in three altitude regions: high

130

atmosphere (HA) made of layers, mid-atmosphere (MA) made of voxels of any size, and

131

bottom atmosphere (BA) in the Earth scene. Earth scene elements are made of facets

132

(triangles), and/or fluid and turbid vegetation voxels. The voxel matrix is introduced to

133

optimize ray tracing.

134 135

2.1 DART atmosphere profiles

136

137

DART simulates the atmosphere as three superimposed volumes: (1) bottom atmosphere (BA) 138

inside the Earth scene voxel matrix, (2) mid-atmosphere (MA) made of voxels, and (3) high

139

7

atmosphere (HA) made of layers (Figure 1). The geometry of MA and HA (i.e., number of 140

layers, layer thickness, voxel size) is either user-defined or analytically computed. The 141

geometry of BA is the same as the Earth scene. The DART atmosphere SQL database stores 142

vertical profiles of atmospheric constituents (i.e., temperature 𝑇

(𝑧) , pressure 𝑃

(𝑧) , 143

number density 𝑁

(𝑧) per gas 𝑚

, relative density 𝜌

(𝑧) of scattering gases to air at 144

standard temperature and pressure and aerosol extinction coefficient profile 𝛼

(𝑧) at 550 145

nm). These profiles are stored at 36 altitude levels (0 to 25 km with 1 km interval, 30 to 60 km 146

with 5 km interval and 3 levels at 70 km, 80 km and 100 km) for:

147

- six standard atmospheres (Anderson et al., 1986): (1) TROPICAL: Tropical (15°N annual 148

average), (2) MIDLATSUM: Mid-Latitude Summer (45°N July), (3) MIDLATWIN: Mid- 149

Latitude Winter (45°N January), (4) SUBARCSUM: Sub-Arctic Summer (60°N July), (5) 150

SUBARCWIN: Sub-Arctic Winter (60°N January), and (6) USSTD76: US Standard 1976.

151

- five aerosol models (Shettle and Fenn, 1979): (1) Rural, (2) Urban, (3) Maritime, (4) 152

Tropospheric and (5) Fog.

153 154

The DART atmosphere SQL database also stores the spectral optical properties of atmospheric 155

constituents (i.e., gas: vertical absorption transmittance 𝑡

(𝜆) per gas 𝑚

, vertical 156

scattering transmittance 𝑡

(𝜆); aerosol: vertical optical depth 𝜏

(𝜆), single scattering 157

albedo 𝜔

(𝜆), asymmetry factors of double Henyey-Greenstein phase function) from 10 to 158

40000 cm

with a spectral resolution of 1 cm

. They were derived from MODTRAN 159

simulations and LOWTRAN source code for the six standard atmospheres and for the five 160

aerosol models per standard atmosphere. The optical properties and vertical profiles of gases 161

and aerosols derived from reanalysis datasets (e.g., ECMWF reanalysis: https://www.ecmwf.int) 162

and measurements (e.g., Aeronet: https://aeronet.gsfc.nasa.gov) can also be imported into the 163

DART atmosphere database.

164

8 165

The atmosphere properties at any altitude z are interpolated by the multi-quadric RBF (Radial 166

Basis Function) (Press et al., 2007) using vertical profiles and optical properties in the SQL 167

database. The band (central wavelength 𝜆, bandwidth ∆𝜆) mean optical properties (i.e., vertical 168

absorption transmittance 𝑡

(𝜆) of each gas 𝑚

, gas vertical scattering transmittance 𝑡

(𝜆), 169

aerosol vertical optical depth 𝜏

(𝜆)) are computed (trapezoidal integration) using the database 170

spectral vertical transmittance 𝑡

(𝜆

) , 𝑡

(𝜆

) and optical depth 𝜏

(𝜆

) at 1 cm

171

spectral resolution in the spectral bin ∆𝜆:

172

𝑡

(𝜆) =

()∆(/, (-∆(/,

∆.

, 𝑡

(𝜆) =

()∆(/, (-∆(/,

∆.

𝜏

(𝜆) = ∫

𝜏

(𝜆

) 𝑑𝜆

∆𝜆

(

1

173

In DART flux tracking mode, the extinction coefficient 𝛼 (i.e., total 𝛼

, absorption 𝛼

and 174

scattering 𝛼

extinction coefficient) are constant values per layer

175

Beer's law gives the band vertical transmittance and optical depth computed in Eq.

1

. 176

𝛼

(𝜆) = − ln 9𝑡

(𝜆):

𝑧

− 𝑧

∙ ∫

𝜎

(𝜆) ∙ 𝑁

(𝑧)𝑑𝑧

/-0

∫ 𝜎

(𝜆) ∙ 𝑁

(𝑧)𝑑𝑧

𝛼

(𝜆) = − ln=𝑡

(𝜆)>

𝑧

− 𝑧

∙ ∫

𝜎

(𝜆) ∙ 𝑁

(𝑧)𝑑𝑧

/-0

∫ 𝜎

(𝜆) ∙ 𝑁

(𝑧)𝑑𝑧 = − ln=𝑡

(𝜆)>

𝑧

− 𝑧

∙ ∫

𝜌

(𝑧)𝑑𝑧

/-0

∫ 𝜌

(𝑧)𝑑𝑧

𝛼

(𝜆) = 𝜏

(𝜆)

𝑧

− 𝑧

∙ ∫

𝛼

(𝑧)𝑑𝑧

/-0

∫ 𝛼

(𝑧)𝑑𝑧

(

2

177

The Newton-Cotes integration method is used in Eq.

2

with 10 interpolated equal-distance 178

values per layer (Abramowitz and Stegun, 1948) assuming that the absorption cross-section

179

9

𝜎

(𝜆) of gas 𝑚

only depends on wavelength. The gas scattering cross-section 𝜎

(𝜆) only 180

depends on wavelength and gas composition (Bodhaine et al., 1999). Therefore, 𝜎

(𝜆) ∙ 𝑁

(𝑧), 181

with 𝑁

(𝑧) being the number density of scattering gases at altitude 𝑧, is proportional to the 182

relative density of scattering gases 𝜌

(𝑧).

183 184

Then, the total gas extinction coefficient 𝛼

(𝜆), aerosol absorption 𝛼

(𝜆) and scattering 185

𝛼

(𝜆) extinction coefficient and total extinction coefficient 𝛼

(𝜆) are computed per layer:

186

𝛼

(𝜆) = 𝛼

(𝜆) + 𝛼

(𝜆) , 𝛼

(𝜆) = ∑ 𝛼

(𝜆)

𝛼

(𝜆) = 𝛼

(𝜆) ∙ 91 − 𝜔

(𝜆): , 𝛼

(𝜆) = 𝛼

(𝜆) ∙ 𝜔

(𝜆)

𝛼

(𝜆) = 𝛼

(𝜆) + 𝛼

(𝜆)

(

3

187

As explained in section 4, in order to improve the modelling of thermal emission, we adapted 188

the continuous optical depth profiles per atmosphere layer computed in the DART LiDAR 189

mode (Gastellu-Etchegorry et al., 2015). Hence, 𝛼

(𝜆, ℎ) and 𝜏

(𝜆, ℎ)= ∫

𝛼

(𝜆, ℎ)𝑑ℎ are 190

continuous functions per atmosphere layer

defined as 191

𝜏

(𝜆, ℎ)=𝐴

(𝜆)∙ℎ

+𝐵

(𝜆)∙ℎ

+𝐶

(𝜆)∙ℎ+𝐷

(𝜆) and 𝛼

(𝜆, ℎ)=-3𝐴

(𝜆)∙ℎ

-2𝐵

(𝜆)∙ℎ-𝐶

(𝜆) , 192

where h is the relative altitude in layer j, with ℎ=0 at the bottom of layer j and ℎ=Δ𝑧

193

of layer

𝜏(𝜆, 0)=∆𝜏

(𝜆), 𝜏=𝜆, Δ𝑧

>=0, 194

𝛼

(𝜆, 0)=𝛼

(𝜆), 𝛼

=𝜆, ∆𝑧

>=𝛼

(𝜆) derives the coefficients 𝐴

(𝜆), 𝐵

(𝜆), 𝐶

(𝜆) and 𝐷

(𝜆) 195

(Eq.

4

).

196

𝐴

(𝜆) =

2 (.)3B_1/² (.)CD9_/

D9_/³

, 𝐵

(𝜆) =

,3B_1/-0² (.)6B_1/² (.)

5D9_/ (

4

10

𝐶

(𝜆) = −𝛼

(𝜆), 𝐷

(𝜆) = ∆𝜏

(𝜆)

197

Figure 2. DART horizontally homogeneous atmosphere layer with layer thickness Δ𝑧_!. The upper and

198

lower boundary parameters are marked.

199 200

2.2 Layer thermal emission

201

202

Thermal emitted vector source (Eq.

5

) of a DART atmosphere layer j (horizontal surface ∆𝑆, 203

thickness Δ𝑧

) per direction vector Ω(ΔΩ) (zenith angle 𝜃, azimuth angle 𝜑) is computed using 204

layer mean temperature 𝑇

and two optical properties: single scattering albedo 𝜔

(𝜆)=

/2(.)

and 205

extinction coefficient 𝛼

(𝜆)=𝛼

(𝜆) + 𝛼

(𝜆) , with 𝛼

(𝜆) = 𝛼

(𝜆) + 𝛼

(𝜆) the total 206

absorption extinction coefficient and 𝛼

(𝜆) = 𝛼

(𝜆) + 𝛼

(𝜆) the total scattering extinction 207

coefficient.

208

𝑊

(Ω, 𝜆) = Q 𝐿

=𝑇

, 𝜆> ∙ 𝛼

(𝜆) ∙ ∆𝑆 ∙ ∆Ω ∙ 𝑒

𝑑𝑧

∆9_/

;

= 91 − 𝜔

(𝜆): ∙ 𝐿

=𝑇

, 𝜆> ∙ T1 − 𝑒

U ∙ 𝜇 ∙ ∆𝑆 ∙ ∆Ω

(

5

where 𝐿

=𝑇

, 𝜆> (unit: 𝑊/m

/sr/𝜇m) is the spectrally averaged Planck function at layer mean 209

temperature 𝑇

. ∆𝜏

is the optical depth of the atmosphere layer j. 𝜇 = cos (𝜃).

210

211

11

2.3 RT in the Earth-atmosphere system

212

213

RT modelling in the Earth-atmosphere system involves five major steps (Figure 3): (1) Sun 214

illumination and atmosphere thermal emission; (2) Earth surface RT; (3) Earth-atmosphere 215

radiative coupling and atmosphere backscattering; (4) Earth surface RT of the backscattered 216

radiation; (5) Transfer of bottom of atmosphere (BOA) radiation to any altitude, including TOA 217

(Grau and Gastellu-Etchegorry, 2013).

218

219

Figure 3.Major steps for modelling the RT in the Earth-atmosphere system. Red colour indicates

220

thermal emission (steps 1 and 2), orange colour indicates solar incident radiation, and yellow

221

colour indicates thermal and/or solar radiation that is scattered.

222 223

3. Initial DART atmospheric RT modelling accuracy

224

225

DART atmospheric RT modelling accuracy was assessed using MODTRAN-5 since it is one 226

of the most accurate atmospheric RT models, with transmittance accuracy ±0.005, radiance 227

accuracy ±2% and thermal brightness temperature (BT) accuracy better than 1 K (Berk et al., 228

2008, 2005, 1987). In the short waves (i.e., [0.4 𝜇m, 3.0 𝜇m]), with ground albedo 0.5, TOA 229

nadir reflectance absolute difference between DART and MODTRAN-5 was less than 0.004,

230

12

which meets MODTRAN-5 accuracy (Gastellu-Etchegorry et al., 2017; Grau and Gastellu- 231

Etchegorry, 2013). Here, the comparison is extended to TIR region from 3.5 𝜇m to 20 𝜇m, 232

using DART version 5.7.3, hereafter called "initial DART". To analyse pure gas atmosphere 233

emission, MODTRAN-5 and DART were run with thermal emission mode, no aerosol, the 234

Earth skin temperature is 0 K, surface albedo is 0, and the atmosphere layer depth is equal to 1 235

km from 0 to 25 km, and 5 km from 30 km to 100 km. Figure 4 shows DART and MODTRAN- 236

5 TOA and BOA TIR radiance spectra and the residuals, for four standard atmospheres 237

(USSTD76, TROPICAL, MIDLATSUM, SUBARCWIN). The mean absolute error (MAE) 238

and mean absolute relative error (MARE) instead of the root-mean-square-error (RMSE) are 239

used to quantify DART and MODTRAN-5 differences since they bring a more unambiguous 240

information (Willmott and Matsuura, 2005). For a variable 𝑋(𝑞) (e.g., BT, radiance) at band 𝑞 241

and Q spectral bands:

242

MAE = 1

𝑄 . d|𝑋

(𝑞) − 𝑋

(𝑞)|

O

PQ8

MARE = 1

𝑄 . d |𝑋

(𝑞) − 𝑋

(𝑞)|

𝑋

(𝑞) ∙ 100

O

PQ8

%

(

6

243

Note that the error defined in Eq.

6

is relative to MODTRAN, one should take into account 244

the accuracy of MODTRAN in practical application.

245

13 a)

b)

c)

14 d)

Figure 4. Initial DART and MODTRAN-5 TOA / BOA TIR radiance in [3.5 𝜇m, 20 𝜇m] region for

246

USSTD76 (a), TROPICAL (b), MIDLATSUM (c) and SUBARCWIN (d) atmospheres. 1

247

cm^-1 spectral resolution. DART and MODTRAN-5 configurations are detailed in section 3.

248 249

Table 1 summarizes the "Initial DART – MODTRAN-5" MAEs for BT and MAREs for 250

radiance, for six standard atmospheres. For most standard atmospheres, BT MAEs are larger 251

than 2.0 K and 1.5 K for TOA and BOA radiance spectra, respectively. The maximal BT MAE 252

occur for the TROPICAL atmosphere at TOA level with Max (BT MAE) = 4.7 K.

253 254

Table 1. TOA and BOA BT MAE and radiance MARE of initial DART in [3.5 𝜇m, 20 𝜇m] region for

255

six standard atmospheres. MODTRAN-5 results are the reference.

256

Atmosphere

BT MAE Radiance MARE

TOA (K) BOA (K) TOA (%) BOA (%)

USSTD76 3.1 K 2.7 K 12.6 8.6

TROPICAL 4.7 K 2.4 K 18.9 6.8

MIDLATSUM 3.8 K 2.1 K 15.0 6.1

15

MIDLATWIN 2.3 K 1.8 K 10.1 6.3

SUMARCSUM 2.9 K 2.3 K 11.5 7.1

SUMARCWIN 1.8 K 1.3 K 8.0 5.2

Average 3.1 K 2.1 K 12.7 6.7

257

DART and MODTRAN-5 TOA / BOA radiance values diverge more in absorbing spectral 258

regions than in non-absorbing spectral regions. For example, Table 2 shows that very large BT 259

differences, up to 7 K, occur in the four absorbing spectral regions: ABS1 ([3.5 𝜇m, 4.5 𝜇m]), 260

ABS2 ([6 𝜇m, 7 𝜇m]), ABS3 ([9 𝜇m, 10 𝜇m]) and ABS4 ([14 𝜇m, 16 𝜇m]) for the USSTD76 261

atmosphere.

262 263

Table 2. TOA and BOA BT MAE and radiance MARE of initial DART in four TIR absorbing bands

264

(ABS1, ABS2, ABS3, ABS4) for the USSTD76 atmosphere. MODTRAN-5 results are the

265

reference.

266 267

268

DART accuracy was also analysed in relation to three EO satellite missions (Trishna, Sentinel 269

3, Landsat 8). Table 3 shows the BT difference (DIFF) of their TIR bands, for the USSTD76 270

Absorption band

Central wavelength

Bandwidth

BT MAE Radiance MARE

TOA (K) BOA (K) TOA (%) BOA (%)

ABS1 4.0 𝜇m 1.0 𝜇m 2.0 K 1.7 K 12.7 8.3

ABS2 6.5 𝜇m 1.0 𝜇m 5.4 K 3.0 K 25.1 7.9

ABS3 9.5 𝜇m 1.0 𝜇m 2.2 K 3.5 K 7.7 11.0

ABS4 15.0 𝜇m 2.0 𝜇m 7.0 K 3.4 K 13.8 4.0

16

atmosphere. Band BTs are computed by inverting Planck function using band mean radiance 271

and band central wavelength. The resulting DART BT DIFFs greatly exceed the satellite sensor 272

sensitivity (Table 3) and also the accuracy usually required for LST applications (i.e., 1 K).

273 274

Three already mentioned DART approximations can explain these large differences: (1) 275

Neglect of gas absorption cross-section dependence on pressure and temperature. Indeed, due 276

to the Doppler and Lorentz broadening, absorption cross-sections vary with pressure and 277

temperature, and consequently with altitude; (2) Transmittance computation with Beer's law 278

and band mean optical properties. Indeed, Beer's law is less correct if the absorption cross- 279

section varies strongly within the spectral bin; (3) Computation of layer thermal emission with 280

the layer mean temperature 𝑇

, which is only suited for optically thin layers. Therefore, with 281

the objective that DART accuracy meets the requirements of TIR EO satellite missions and 282

LST applications while using Beer's law, two major modelling improvements have been made:

283

(1) account of the vertical variation of gas absorption cross-section, and (2) accurate 284

computation of thermal emission per layer.

285 286

Table 3. TOA BT DIFF of initial DART in the TIR bands of three EO satellite missions for the

287

USSTD76 atmosphere. MODTRAN-5 results are the reference.

288

Satellite Launch date Organization

Central wavelength

Bandwidth

Sensitivity (NeDT)

DIFF

Trishna Foreseen CNES+ISRO 8.6 𝜇m 0.35 𝜇m 0.3 K@300 K 0.65 K

2024-2025 9.1 𝜇m 0.35 𝜇m 0.3 K@300 K 1.57 K

10.3 𝜇m 1.0 𝜇m 0.3 K@300 K 2.60 K

17 289

4. Improvement to TIR RT modelling

290

4.1 Equivalent absorption cross-section database

291

4.1.1 Equivalent absorption cross-section

292

293

As stated above, the initial DART neglects the dependence of gas absorption cross-sections 294

with pressure and temperature. We improved this situation by introducing vertical profiles of 295

equivalent absorption cross-section 𝜎

(𝜆, 𝑧, Δ𝐿) (Eq.

7

) for five most absorbing gases (H

O, 296

CO

, O

, CH

, N

O). 𝜎

is the exact band mean absorption cross-section if Beer's law is 297

obeyed.

298

𝜎

(𝜆, 𝑧, 𝛥𝐿) = − ln 9𝑡

(𝜆, 𝑧, 𝛥𝐿):

𝑁

(𝑧) ∙ 𝛥𝐿

7

299

with 𝑡

(𝜆, 𝑧, Δ𝐿) the path absorption transmittance of gas 𝑚

at altitude

300

along path segment Δ𝐿. 𝑁

(𝑧) is the number density of gas 𝑚

at altitude z.

301 302

11.5 𝜇m 1.0 𝜇m 0.3 K@300 K 1.50 K Landsat 8 2013 NASA 10.9 𝜇m 0.6 𝜇m 0.4 K@300 K 1.97 K 12.0 𝜇m 1.0 𝜇m 0.4 K@300 K 1.73 K Sentinel 3 2016 ESA 3.74 𝜇m 0.38 𝜇m 0.08 K@270 K 0.10 K 10.95 𝜇m 0.9 𝜇m 0.05 K@270 K 1.88 K 12.0 𝜇m 1.0 𝜇m 0.05 K@270 K 1.73 K

18

To compute 𝜎

(𝜆, 𝑧, 𝛥𝐿) (Eq.

7

), MODTRAN 1 cm

resolution absorption transmittances 303

𝑡

(𝜆, 𝑧, Δ𝐿) were simulated per gas 𝑚

(H

O, CO

, O

, CH

, N

O) for identical horizontal 304

paths Δ𝐿 at 36 altitudes z: 26 altitudes from 0 km up to 25 km with a step of 1 km, 7 altitudes 305

from 25 km up to 60 km with a step of 5 km, and 3 altitudes at 70 km, 80 km and 100 km. The 306

gas number density 𝑁

(𝑧) is derived from MODTRAN tape6 file (Berk et al., 2008).

307 308

Figure 5.a and Figure 5.c show the CO

vertical profiles of 𝜎

(𝜆, 𝑧, Δ𝐿) from 0 km altitude up 309

to 25 km for 13.4 𝜇m and up to 15 km for 13.1 𝜇m, with Δ𝐿 = 1 km, 2 km, 5 km and 10 km.

310

𝜎

(𝜆, 𝑧, Δ𝐿) profiles depend on Δ𝐿 if Beer's law is not obeyed (Figure 5.a) and do not depend 311

on Δ𝐿 if Beer's law is obeyed (Figure 5.c). Figure 5.b, d shows that the relative vertical 312

distribution of equivalent absorption cross-section 𝜎

(𝜆, 𝑧, Δ𝐿)=

S_"!^& (.,;,<T)

(profiles scaled 313

by 𝜎

(𝜆, 0, Δ𝐿) at z = 0) is almost independent of Δ𝐿, even if Beer's law is not obeyed, with 314

slightly variations depending on the absorption feature, temperature and pressure. It explains 315

that DART TIR radiance computed with 𝜎

(𝜆, 𝑧, 𝛥𝐿) only slightly varies with Δ𝐿. Results in 316

section 5 show that the optimal Δ𝐿 is 7 km. Hereafter, 𝜎

(𝜆, 𝑧) stands for 𝜎

(𝜆, 𝑧, Δ𝐿 = 317

7 km).

318 319

Note that 𝜎

(𝜆, 𝑧) is underestimated if the minus logarithm of total transmittance reaches the 320

limit 100 in MODTRAN (i.e., extreme absorbing bands). It can occur at low altitude. Then, the 321

low atmosphere tends to behave as a black body, and the vertical distribution of absorption 322

extinction coefficient within the low atmosphere has little impact on TIR radiance.

323

324

19

a) b)

c) d)

Figure 5. CO2 equivalent absorption cross-section 𝜎_"^#_!(𝜆, 𝑧, 𝛥𝐿) (a, c) and rescaled equivalent

325

absorption cross-section 𝜎_"^#(∗)_! (𝜆, 𝑧, Δ𝐿) (b, d) at 13.4 𝜇m (a, b) and 13.1 𝜇m (c, d), in 1 cm^-1

326

spectral bin, for 4 identical horizontal paths ( Δ𝐿 = 1 km, 2 km, 5 km, 10 km) at altitudes up

327

to 25 km for 13.4 𝜇m and up to 15 km for 13.1 𝜇m, in the USSTD76 atmosphere.

328 329

4.1.2 Creation of SQL database

330

331

The equivalent absorption cross-section profiles 𝜎

(𝜆, 𝑧) were computed per gas 𝑚

for the 332

six standard atmospheres, from 10 to 3500 cm

at 1 cm

spectral resolution. To ease data access 333

and management, they were stored in a SQL database per spectral band, per gas 𝑚

and per 334

standard atmosphere. Three criteria were used to select the spectral bands of interest: (1) 335

wavelength larger than 3 𝜇m; (2) absorbing gases; (3) absorbing spectral regions.

336

20 337

The spectral region over 3 𝜇m is chosen because the vertical variation of absorption cross- 338

section impacts much more the TIR region than short waves. Also, since non-absorbing gases 339

have a negligible impact on the TOA / BOA radiance, 𝜎

(𝜆, 𝑧) is stored only for the five most 340

absorbing gases (H

O, CO

, O

, CH

, N

O) and for the absorbing spectral regions of these gases.

341

This trade-off allows one to get accurate TIR RT modelling without increasing too much the 342

DART code complexity and computer time. In non-absorbing bands, MODTRAN absorption 343

transmittance 𝑡

(𝜆, 𝑧) of gas 𝑚

at any altitude z along the horizontal path is very close to 1 344

(e.g., 0.99995), which implies that the computed equivalent absorption cross-section is either 345

zero or inaccurate. Therefore, we selected absorbing regions per gas 𝑚

. For that, a specific 346

altitude 𝑍

is defined per gas 𝑚

such that under this altitude over 98% of gas 𝑚

is present.

347

𝑍

is 8 km, 𝑍

is 25 km, 𝑍

is 40 km, 𝑍

and 𝑍

are both 23 km. If the sum of 348

equivalent optical depth - ln 9𝑡

(𝜆, 𝑧): for altitude 𝑧 > 𝑍

is not negligible (Eq.

8

), the 349

spectral band is considered as an absorbing band for gas 𝑚

, and the 𝜎

(𝜆, 𝑧) profile is stored.

350

d − ln 9𝑡

(𝜆, 𝑧): > 𝜀

9WX_"!^∗ (

8

351

where the threshold 𝜀 corresponds to MODTRAN precision. Any line-of-sight with equivalent 352

optical depth - ln 9𝑡

(𝜆, 𝑧): smaller than 𝜀 is considered as transparent.

353 354

4.1.3 Improved absorption extinction coefficient profile

355

356

Eq.

9

indicates how the initial gas absorption extinction coefficient Eq.

2

was improved 357

using the pressure- and temperature-dependent equivalent absorption cross-section.

358

21

𝛼_!,"^# _!(𝜆) =

⎩⎪

⎨

⎪⎧ − ln 5𝑡^"#!(𝜆)7

𝑧_!− 𝑧_!() ∙∫_,^," 𝜎_"^#_!(𝜆, 𝑧) ∙ 𝑁_"!,*+(𝑧)𝑑𝑧

"#$

∫ 𝜎_.^- _"^#_!(𝜆, 𝑧) ∙ 𝑁_"!,*+(𝑧)𝑑𝑧 , 𝑚_/= H₀O, CO₀, O₁, CH₁, N₀O

− ln 5𝑡_"^#_!(𝜆)7

𝑧_!− 𝑧_!() ∙∫_,^," 𝜎_"^#_!(𝜆) ∙ 𝑁_"!,*+(𝑧)𝑑𝑧

"#$

∫ 𝜎_.^- _"^#_!(𝜆) ∙ 𝑁_"!,*+(𝑧)𝑑𝑧 , 𝑚_/ ≠ H₀O, CO₀, O₁, CH₁, N₀O (

9

359

Figure

shows vertical profiles of the new absorption extinction coefficients in the USSTD76 360

atmosphere at four spectral bands. Compared to the initial absorption extinction coefficients, 361

they tend to be larger at lower altitudes and smaller at higher altitudes. This is consistent with 362

the stronger absorption behaviour of bottom atmosphere. O

absorption explains the local 363

maximum (» 20 km) of the initial and new absorption extinction coefficients at 10 𝜇m.

364

a) b)

c) d)

365

atmosphere. a) 10.0𝜇m. b) 13.0𝜇m. c) 15.4𝜇m. d) 19.6𝜇m. Spectral bin is 1cm^-1.

366

22 367

4.2 Layer thermal emission

368

4.2.1 A double-layer method

369

370

The initial thermal emission method (Eq.

5

) is less correct for optically thick atmosphere layer 371

(i.e., ∆𝜏

≫1 ). For example, if lower boundary temperature is larger than layer mean 372

temperature and if ∆𝜏

≫1, Eq.

5

tends to underestimate downward thermal vector sources 373

𝑊

(Ω, 𝜆). Hence, a double-layer method was first designed (Eq.

10

): half a layer emits with 374

Planck function 𝐿

(𝑇

, 𝜆), and the other half emits with Planck function 𝐿

=𝑇

, 𝜆>.

375

𝑊

(Ω, 𝜆) = 91 − 𝜔

(𝜆): ∙ T𝐿

(𝑇

, 𝜆)𝑒

∆>_/

5F

+ 𝐿

=𝑇

, 𝜆>U T1 − 𝑒

∆>_/

5F

U ∙ 𝜇 ∙ ∆𝑆 ∙ ∆Ω

𝑊

(Ω, 𝜆) = 91 − 𝜔

(𝜆): ∙ T𝐿

(𝑇

, 𝜆) + 𝐿

=𝑇

, 𝜆>𝑒

∆>_/

5F

U T1 − 𝑒

∆>_/

5F

U ∙ 𝜇 ∙ ∆𝑆 ∙ ∆Ω

(

10

376

In order to get 𝑇

and 𝑇

, four equations associated to schematic configurations must be verified:

377

1) Blackbody (∆𝜏

≫ 1 and 𝜔

(𝜆) ≈ 0) 378

𝑊

(Ω, 𝜆) = 𝐿

=𝑇

, 𝜆> ∙ 𝜇 ∙ ∆𝑆 ∙ ∆Ω 𝑊

(Ω, 𝜆) = 𝐿

=𝑇

, 𝜆> ∙ 𝜇 ∙ ∆𝑆 ∙ ∆Ω

(

11

379

with 𝑇

and 𝑇

respectively the upper and lower boundary temperature of layer j.

380

2) Isothermal (𝑇

= 𝑇

= 𝑇

) 381

𝑊

(Ω, 𝜆) = 91 − 𝜔

(𝜆): ∙ 𝐿

=𝑇

, 𝜆> ∙ T1 − 𝑒

U ∙ 𝜇 ∙ ∆𝑆 ∙ ∆Ω

12

23

𝑊

(Ω, 𝜆) = 91 − 𝜔

(𝜆): ∙ 𝐿

=𝑇

, 𝜆> ∙ T1 − 𝑒

∆>_/

F

U ∙ 𝜇 ∙ ∆𝑆 ∙ ∆Ω

382

Eq.

11

and

12

lead to 𝑇

=𝑇

and 𝑇

=𝑇

. The resulting upward and downward vector 383

sources are:

384

𝑊

(Ω, 𝜆) = 91 − 𝜔

(𝜆): ∙ T𝐿

=𝑇

, 𝜆>𝑒

∆>_/

5F

+ 𝐿

=𝑇

, 𝜆>U T1 − 𝑒

∆>_/

5F

U ∙ 𝜇 ∙ ∆𝑆 ∙ ∆Ω

𝑊

(Ω, 𝜆) = 91 − 𝜔

(𝜆): ∙ T𝐿

=𝑇

, 𝜆> + 𝐿

=𝑇

, 𝜆>𝑒

∆>_/

5F

U T1 − 𝑒

∆>_/

5F

U ∙ 𝜇 ∙ ∆𝑆 ∙ ∆Ω

13

385

4.2.2 Thermal emission with virtual sub-layers

386

387

Same as the "linear-in-optical depth" assumption (Clough et al., 1992), the double-layer method 388

(Eq.

13

) is less correct if the temperature and absorption extinction coefficient gradient within 389

a layer is large (Wiscombe, 1976). Hence, there is a need to take into account the vertical 390

distribution of temperature and optical depth in each atmosphere layer. For that, each DART 391

atmosphere layer 𝑗 (𝑗

[1, 𝐽], 𝑗 = 1 for the bottom layer) is virtually divided into 𝑘

sub-layers 392

∆𝜏

(𝜆) = ∫

𝑑𝜏

6-0(.)

with 𝑘

[1, 𝑘

] (Figure 7). The optical depth of the bottom and top planes 393

of layer 𝑗 are noted 𝜏

(𝜆) and 𝜏

(𝜆), respectively, with 𝜏

(𝜆) = 0 and 𝜏

(𝜆) = Δ𝜏

(𝜆). The 394

terms ∆𝜏

(𝜆), 𝜏

(𝜆), 𝜏

(𝜆), 𝜏

(𝜆) and 𝜏

(𝜆) are computed using analytical expressions of 395

layer optical depth that was already implemented in DART for LiDAR mode (i.e., 𝜏

(ℎ, 𝜆) = 396

𝐴

(𝜆) ∙ ℎ

+ 𝐵

(𝜆) ∙ ℎ

+ 𝐶

(𝜆) ∙ ℎ + 𝐷

(𝜆)). Temperature profile 𝑇

(ℎ) is written as a linear 397

approximation: 𝑇

(ℎ) =

+

/

∙ ℎ.

398

24 399

Figure 7. DART atmosphere is made of 𝐽 layers, with layer optical depth ∆𝜏_! and layer thickness ∆𝑧_!.

400

Each layer 𝑗 is virtually divided into 𝑘_! sub-layers, with sub-layer thickness ∆𝑧₅ =^∆,₅^"

" and

401

sub-layer optical depth ∆𝜏₅, with upper and lower boundary temperature 𝑇₅ and 𝑇₅₍₎,

402

respectively.

403 404

The expressions of 𝑊

(Ω, 𝜆) and 𝑊

(Ω, 𝜆) are computed by summing up all contributions of 405

virtual sub-layer thermal emission using Eq.

13

. 406

𝑊_!^↑(Ω, 𝜆) = (1 − 𝜔_!(𝜆), - .𝐿#(𝑇_$%&, 𝜆)𝑒^%∆()*!+ 𝐿_#(𝑇_$, 𝜆)3 .1 − 𝑒^%∆()*!3 𝑒^%(*!

$_"

$+&

∙𝜇∙∆𝑆∙∆Ω

𝑊_!^↓(Ω, 𝜆) = (1 − 𝜔_!(𝜆), - .𝐿#(𝑇_$%&, 𝜆) + 𝐿_#(𝑇_$, 𝜆)𝑒^%∆()*!3 .1 − 𝑒^%∆()*!3

$_"

$+&

𝑒^{%(("#$%(* !#$)}∙𝜇∙∆𝑆∙ ∆Ω

(

14

407

The optimal number 𝑘

of sub-layers in Eq.

14

was assessed by computing the upward vector 408

sources (𝜔

=0, 𝜇=1,

) of a hot (~300 K) and cold (~200 K) atmosphere 409

layer (∆𝑧

=1 km) with small (0.2) and large (0.8) transmittance 𝑒

, for 𝑘

from 1 to 10, from 410

3 𝜇m to 20 𝜇m. Bottom layer parameters are: 𝑇

=𝑇

, 𝛼

(0, 𝜆)=𝛼

, 𝜏

(0, 𝜆)=∆𝜏

; upper layer 411

parameters are: 𝑇

=𝑇

, 𝛼

(∆𝑧

, 𝜆)=𝛼

∙ 𝑒

, 𝜏

=∆𝑧

, 𝜆>=0, with H = 8.4 km the usual scale

412

25

height of major gases. Sub-layer boundary temperature and optical depth are computed as 413

described at the beginning of this section. Here, the reference is the vector source 𝑊

(

𝜆) 414

computed with 𝑘

= 1000. Figure 8 shows that MARE for 𝑘

= 1 can reach 7% and that 𝑘

= 5 415

gives accurate source vectors for most atmospheric conditions. Note that for atmospheric 416

conditions less extreme than these in this test, the double-layer method usually gives better 417

results.

418 419

a) b)

c) d)

Figure 8. Difference of DART double-layer upward vector source compared with the reference, for

420

various numbers of sub-layers. 𝑇⁸ and 𝑇⁹ are respectively the upper and lower boundary

421

temperatures. TRANS represents the layer transmittance. MARE is marked in the legend.

422

423

26

5. Results and discussion

424

5.1 Results

425

426

The combined introduction of the equivalent absorption cross-section and double-layer thermal 427

emission method greatly improves the accuracy of DART TIR radiance as illustrated below.

428

5.1.1 Optimal path length ∆𝑳

429

430

Section 4 stresses that the path length ∆𝐿 used to compute the equivalent absorption cross- 431

section can slightly impact the TIR radiance. Therefore, we investigated 10 equivalent 432

absorption cross-section databases with ∆𝐿 from 1 km to 10 km, with 1 km interval, in order to 433

determine the optimal ∆𝐿 . TOA and BOA radiance spectra over [3.5 𝜇m, 20 𝜇m] were 434

simulated with these databases for the six standard atmospheres. Table 4 shows the 435

corresponding average MAE of TOA and BOA BT of six standard atmospheres for the 436

equivalent absorption cross-section databases with ∆𝐿 from 4 km to 9 km. We chose ∆𝐿=7 km 437

since it gives the best results compared to MODTRAN-5. Note that ∆𝐿 = 6 km, 8 km, and 9 438

km can also give good results with average BT MAE less than 0.7 K. Hereafter, all the DART 439

simulations use the "∆𝐿=7 km" absorption cross-section database.

440 441

Table 4. Average MAE of TOA and BOA BT over [3.5 𝜇m, 20 𝜇m] region of six standard atmospheres,

442

with path lengths ∆𝐿 from 4 km to 9 km. MODTRAN-5 results are the reference.

443

∆𝐿 4 km 5 km 6 km 7 km 8 km 9 km

AVG BT MAE (K)

0.7489 0.7157 0.6828 0.6728 0.6731 0.6805

444